Problem: The line $y = \frac{1}{2} x + 4$ is paramterized by the form
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -7 \\ s \end{pmatrix} + t \begin{pmatrix} l \\ -5 \end{pmatrix}.\]Enter the ordered pair $(s,l).$
Explanation: Taking $t = 0,$ we find $\begin{pmatrix} -7 \\ s \end{pmatrix}$ lies on the line.  Then
\[s = \frac{1}{2} (-7) + 4 = \frac{1}{2}.\]Taking $t = 1,$ we get
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -7 \\ 1/2 \end{pmatrix} + \begin{pmatrix} l \\ -5 \end{pmatrix} = \begin{pmatrix} -7 + l \\ -9/2 \end{pmatrix}.\]Then
\[-\frac{9}{2} = \frac{1}{2} (-7 + l) + 4.\]Solving for $l,$ we find $l = -10.$

Hence, $(r,k) = \boxed{\left( \frac{1}{2}, -10 \right)}.$